Chapter 4

4.1Thickness of Plane Beds

Thickness has multiple definitions, the choice of which depends on the purpose and thedata available (Fig. 4.1). The true stratigraphic thickness (TST) is always the distancebetween the top and base of a unit measured perpendicular to the top. In a completely ex-posed outcrop, bed thickness can be measured directly across the bed. In a well that is per-pendicular to bedding, the measured thickness in the well (MD) is the true stratigraphicthickness. Commonly, however, thicknesses must be determined from oblique traversesacross beds or from wells that are not perpendicular to the bed boundaries. The measuredthickness in a vertical well (or along a vertical traverse) is the true vertical thickness (TVT).The measured thickness in any other direction is here termed a slant thickness. A “thick-ness” measurement that is easily derived from well data is the TVD or true vertical depththickness, and is the difference in elevation between the top and base of a unit in a welllog. This “thickness” is more related to the orientation of the well, however, and for a hori-zontal traverse or a horizontal well, the TVD is zero. In the following sections the true strati-graphic thickness is found first and then other thickness determined from it, as needed.

4.1.1Universal Thickness Equation

The stratigraphic thickness can be determined from a single equation (Hobson 1942;Charlesworth and Kilby 1981) based on the angle between the direction of the thick-

Thickness Measurements and Thickness Maps

Fig. 4.1.Vertical cross section in thedip direction showing mea-sures of thickness: MD: mea-sured distance on well log ortraverse; t: true stratigraphicthickness = TST; tv: true verti-cal thickness = TVT; ts: slantthickness; TVD: true verticaldepth thickness = difference inz coordinates between top andbase of unit. (Standard well-logterminology from Robert L.Brown of Shell Oil Co.)

90 Chapter 4 · Thickness Measurements and Thickness Maps

ness measurement and the pole to bedding (Fig. 4.2). The method is convenient fordata from a well or a map. It is here called the universal thickness equation because italways works, regardless of the direction of the measurement or the dip of the bed.The advantage of this method is that the thickness is given by a single, simple equa-tion, eliminating most potential sources of error. The disadvantage for hand calcula-tion is the need to determine an angle in three dimensions, although this is easily donewith a stereogram or spreadsheet. The apparent thickness is measured along thedirection L. In the plane defined by the line of measurement and the pole to bedding

t = L cos ρ , (4.1)

where t = the true stratigraphic thickness and L is the straight-line length between thetop and base of the unit (MD), measured along a well or between two points on a map.The angle ρ = the angle between L and the pole to the bed. If the acute angle is usedin the equation, the thickness is positive. If the obtuse angle is used the thickness will becorrect in magnitude but negative in sign; taking the absolute value gives the correct resultfor either possibility. If L is vertical, then the angle ρ = δ, the dip of the bed. If the truethickness is known, then the vertical thickness can be found by rewriting Eq. 4.1 as

tv = t / cos δ , (4.2)

where tv = vertical thickness, t = true thickness, and δ = true dip.If L is not vertical, its inclination must be determined. For a well, a directional survey

will give the azimuth of the deviation direction and the amount of the deviation; thelatter which may be reported as the kickout angle, the angle up from the vertical. Al-ternatively, the deviation of a well may be given as the xyz coordinates of points alongthe well bore. The angle ρ can be found graphically or analytically, as described in thefollowing two sections.

4.1.1.1Angle between Two Lines, Stereogram

To find the angle ρ with an equal-area stereogram, on an overlay, plot the point repre-senting the pole to bedding by marking the trend of the dip on the overlay, rotating theoverlay to bring this mark to the east-west axis, and counting inward from the outercircle (the zero-dip circle) the amount of the dip plus 90°, and mark the point. Return

Fig. 4.2.Data needed to determinethickness of a unit from theuniversal-thickness equation(Eq. 4.1)

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the overlay to its original position. Plot the line of measurement (or well bore) by simi-larly marking the trend of the measurement on the outer circle, bringing the mark tothe east-west axis and measuring the dip inward from the outer circle if given as aplunge, or outward from the center of the graph if given as a hade or kickout angle.Rotate the overlay until the two points fall on the same great circle (Fig. 4.3). The angle ρis measured along the great circle between the two points.

As an example of the thickness calculation based on the universal thickness equa-tion, find the true thickness of a bed that is L = 10 m thick in a well. The well hades 10°to 310° and the bed dip vector is 20, 015. Plot the bed on the stereogram and find itspole. Then plot the well and measure the angle between the two lines (ρ = 27°). Equa-tion 4.1 gives t = 8.9 m.

4.1.1.2Angle between Two Lines, Analytical

Method 1. Using bed dip vector and well dip vectorTo find the angle between the bed pole and the well, both given as bearing and plunge,substitute well dip vector (Eq. 12.3) and the bed pole from the dip vector (Eq. 12.13)into the equation for the angle between two vectors (Eq. 12.25) to obtain

ρ = cos–1 ρ = –cos δw sin θw sin δb sin θb– cos δw cos θw sin δb cos θb + sin δw cos δb , (4.3)

where ρ = angle between bed pole and fault dip vector, δw = dip of well, θw = azimuthof well dip, δb = dip of bed, θb = azimuth of bed dip.

Fig. 4.3.Equal-area stereogram show-ing the angle between twolines. Angle ρ is the greatcircle distance between thepoints giving the pole to bed-ding and the orientation ofthe apparent thickness mea-surement. Lower-hemisphereprojection

4.1 · Thickness of Plane Beds

92 Chapter 4 · Thickness Measurements and Thickness Maps

Method 2. Using bed dip vector and points on top and base of unitIf the locations of the unit boundaries are given by their xyz coordinates, and the ori-entation of bedding by its azimuth and dip, the angle ρ required in Eq. 4.1 may befound by substituting Eq. 12.9 into Eq. 12.25:

ρ = cos–1 {[(x1 – x2) / L] sin θb sin δb + [(y1 – y2) / L] cos θb sin δb+ [(z2 – z1) / L] cos δb} , (4.4)

where θb and δb =, respectively, the azimuth and dip of the bed dip vector, and L, theapparent length, is

L = [(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2]1/2 . (4.5)

Method 3. Using bed dip vector and line on mapIf the line of the thickness measurement is defined by its map length, h, change inelevation, v, and orientation given by the azimuth, θ, to the lower end of the line, thenthe angle ρ required in Eq. 4.1 may be found by substituting Eq. 12.7 into Eq. 12.25:

ρ = cos–1 {cos δb sin [arctan (v / h)]– sin δb cos [arctan (v / h)] (cos θb cos θ + sin θb sin θ)} , (4.6)

where θb and δb =, respectively, the azimuth and dip of the bed dip vector, and L, theapparent length, is

L = (v2 + h2)1/2 , (4.7)

where v = vertical distance between end points and h = the horizontal distance between endpoints. The angle in Eq. 4.6 must be acute and must be changed to 180 – ρ if it is obtuse.

4.1.2Thickness between Structure Contours

The thickness determined between structure contours is straightforward to compute andgenerally shows much less variability than that determined between individual points on anoutcrop map. This approach provides a more reliable value in situations where the attitudesand contact locations are uncertain on a map. Determining the best-fit structure contoursuses a large amount of data simultaneously to improve the attitude of bedding and thecontact locations. This method requires a structure contour at the top and base of the bed(Figs. 4.4, 4.5a). The width of the unit is always measured in the dip direction, perpen-dicular to the structure contours. If contours at the same elevation can be constructed onthe top and base of the unit, from the geometry of Fig. 4.5b, the thickness of the unit is

t = hc sin δ , (4.8)

where hc = horizontal distance between contours at equal elevations on the top andbase of the unit, t = true thickness, and δ = true dip. If the contours on the top and

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base of the unit are at different elevations (Fig. 4.5c), then the line on the map thatconnects the upper and lower contours has the length L, and the thickness can be cal-culated from Eq. 4.1. Equation 4.1 gives the same result as Eq. 4.8 for the special casewhere L is horizontal.

The vertical thickness can readily be computed by taking the difference in elevationbetween structure contour maps on the top and base of the unit at a given xy point(Fig. 4.5b). Then the true thickness is calculated from Eq. 4.2, rewritten as

t = tv cos δ , (4.9)

where t = true stratigraphic thickness, tv = vertical thickness and δ = dip.

Fig. 4.4. Oblique views of planar unit boundaries cutting a topographic surface. Map view is in Fig. 4.5a.a Upper and lower bed surfaces, view toward north. b View to northeast parallel to bedding showingthickness, t

Fig. 4.5. Thickness measured between structure contours. a Structure contours at 600-ft elevation on thetop and base of a formation; hc is perpendicular to structure contours (3-D views in Fig. 4.4). b Measurementalong a constant elevation on a vertical cross section in the dip direction. c Measurement between pointsof different elevations on a vertical cross section in the dip direction. For explanation of symbols, see text

4.1 · Thickness of Plane Beds

94 Chapter 4 · Thickness Measurements and Thickness Maps

As an example, in Fig. 4.5a, the 600-ft contour has been located on both the top andbase of the Mpm, and so the simplest means of thickness determination is with Eq. 4.8.The value of hc measured from the map is 1 387 ft, and the dip, δ, is 04°, giving a thick-ness of 97 ft.

4.1.3Map-Angle Thickness Equations

The calculation of the thickness of a unit based entirely on map distances and direc-tions results in two equations, depending on whether the topographic slope is in thesame direction as the dip of the unit or the opposite direction to it. For the bed dip andtopographic slope in the same direction, the equation (derived at the end of this chap-ter as Eqs. 4.18 and 4.21) is

t = |h cos α sin δ – v cos δ | , (4.10)

where the notation |…| = the positive value of the expression between the bars. For abed dip and topographic slope that are in opposite directions:

t = h cos α sin δ + v cos δ , (4.11)

where h (Fig. 4.6) = the horizontal distance along a line between the upper and lowercontacts on the map, α = the angle between the measurement line and the dip direc-tion, δ = the true dip of the unit, and v = the elevation difference between the end pointsof the measurement line. If the measurement is in the dip direction, α = 0 and cos α = 1;if the base and top of the unit are both at the same elevation, v = 0.

As an example, determine the thickness of the Pride Mountain Formation (Mpm)indicated by its outcrop width on the map of Fig. 4.7. Along line a on the map, thehorizontal width of the outcrop is h = 1 250 ft, the vertical drop from highest to lowestcontact is v = 100 ft, the attitude of bedding is δ = 07° (the average of the 06 and 08° dipsmapped) at an azimuth θ = 135°, and the angle between the dip direction and themeasurement line is α = 85°. The bed and topography slope in the same direction (strikeof bedding = 225° and the thickness traverse is along azimuth 220°), making Eq. 4.10appropriate. The resulting thickness is 86 ft.

Fig. 4.6.Map data needed to determinethe thickness of a unit fromoutcrop observations using themap-angle thickness Eqs. 4.10and 4.11

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4.1.4Effect of Measurement and Mapping Errors

In the determination of thickness from measurements on a map, there are three sourcesof error, the length and the direction measurements, the dip measurement, and thecontact locations. Errors in length will result from the finite widths of the contactlines at the scale of the map. The minimum possible length error is approximatelyequal to the thickness of the geologic contacts as drawn on the map. For example, thegeologic contacts in Fig. 4.7 are about 20 ft wide. Thus a thickness error of about±20 ft is about the best that could be done on this map. Small errors in contact loca-tions may lead to large errors in the calculated thickness, depending on the measure-ment direction. Uncertainties in the dip amount and dip direction of a degree or twoare to be expected. The effects of error in the dip range from very small if the thick-ness measurement is at a high angle to the bed boundary, to very large if the mea-surement is at a low angle to the boundary.

The effect of errors in bedding attitude and measurement direction on a thicknessdetermination can be estimated using Eq. 4.1 which places the variables in their sim-plest form. The true thickness (t) is directly proportional to the apparent thickness (L)and so the erroneous thickness (te) is directly proportional to the error in the lengthof the apparent thickness measurement. The error measure (te) is normalized in Fig. 4.8to remove the effect of the length scale. The true thickness in Eq. 4.1 is proportionalto the cosine of the angle between the pole to the bed and the line of the thicknessmeasurement (ρ), leading to a non-linear dependence of the error on the size of theangle (Fig. 4.8). The angle error could be in the orientation of the bed, in the orienta-tion of the apparent thickness, or a combination of both. A combined error in ρ of ±5°from the correct value seems like the upper limit for careful field or well measure-ments. An angle of ρ = 0° means that the thickness measurement is perpendicular tobedding; this is the most accurate measurement direction. An angle of ρ = 90° meansthat the measurement direction is parallel to bedding, an impossibility. The sensitivityof a thickness measurement to error in the angle goes up rapidly as ρ increases (Fig. 4.8).The thickness error exceeds ±10% at an angle of 40° and is about 100% at 80 + 5° and

Fig. 4.7.Line of a thickness measure-ment (a) on a geologic mapon a topographic base

4.1 · Thickness of Plane Beds

96 Chapter 4 · Thickness Measurements and Thickness Maps

–35% at 80 – 5°. This means that thickness measurements made at a low angle to theplane of the bed may produce very large errors, with a strong bias toward overestima-tion. At angles between the measurement direction and bedding of 10° or less (ρ = 80°or greater), very large thickness errors will occur with very small orientation errorsand thickness determinations made from maps should be considered suspect.

From Fig. 4.8, a bed having a true thickness of 100 m measured at an angle of 10°to bedding (ρ = 80°) for which the angle is overestimated by 5° (giving ρ = 85°) willyield a thickness of nearly 200 m. The same measurement with a 5° underestimate inthe angle will give a thickness of 65 m. The average thickness for these two measure-ments is 132.5 m, still an overestimate. Thickness measurements made nearly per-pendicular to bedding (ρ = 0°) are rather insensitive to errors in the angle. At ρ = 20°the error is about ±3%; a 100 m thick bed would be measured as being between 97and 103 m thick.

Thickness calculations between two points on a map are very sensitive to the accu-racy of the contact locations and the attitude of bedding, as illustrated by the dataobtained from Fig. 4.9a. The thickness of the Mpm from the seven locations a–g(Table 4.1) ranges from 84 to 230 ft, as calculated from Eq. 4.1. This is an unreasonablylarge variation in thickness at what is nearly a single location at the scale of the map.What is the probability that the thickness variation is due to small measurement er-rors? The bedding azimuth of 4, 125 represents the value determined from the struc-ture-contour map (Fig. 4.9b). The difference in dip from 04° on the structure contourmap to 06° from the field measurement is responsible for a large variation in the cal-culated thickness. For example, at point c, the thickness along a single line is 127 ft fora 04° dip compared to 230 ft for a 06° dip (Table 4.1). However, the substantial differ-ence in calculated thickness along lines a and g is not the result of uncertainty in thedip, but must be attributed to the uncertainty both in the location of the lower contactand in the exact dip direction. Changing the azimuth of the dip from 135 to 127 atlocation a increases the thickness from 84 to 108 ft at a bed dip of 08°; at location g thesame change reduces the thickness from 173 to 159 ft. If we say that the thickness of

Fig. 4.8.Effect of attitude and anglemeasurement errors on thick-ness calculation. Normalizederror in thickness measure-ment, in percent as a functionof the angle ρ (Eq. 4.1) forangle errors of ρ = ±5°. t Truethickness of bed; te erroneousthickness given the angle er-ror; ρ angle between the poleto the bed and direction alongwhich thickness is measured

97

Fig. 4.9. Alternative thickness measurements. a Point-to-point measurement lines a–g. b Measure-ments 1–4 between structure contours

Table 4.1. Thickness data in map area of Fig. 4.9. All thicknesses calculated with Eq. 4.1

the Mpm is the average of the values at locations a–g obtained using the observed dips,then the thickness is 181 ft with a range from 84 to 230 ft, a poorly constrained result.

The thicknesses determined at locations 1–4 (Fig. 4.9b) between the structure con-tours on the top and base of the unit average 108 ft thick and range from 97 to 120 ft

4.1 · Thickness of Plane Beds

98 Chapter 4 · Thickness Measurements and Thickness Maps

(Table 4.1) or 108 ±12 ft based on the whole range of values. The SE lengths (Table 4.1)are measured to a southeasterly position on the base of the Mpm, at the 600-ft contour,which lies directly beneath the 700-ft contour on the top of the unit; the NW measure-ments are from the more northwesterly position of the lower contact. Changing thelocation of the structure contour of the base has only a small effect on the thickness.The average thickness determined from the structure-contour-based measurementsfalls within the range of the point-to-point measurements, but is much smaller thanthe average of the point-to-point measurements, as expected from the behavior of thethickness equation (Fig. 4.8). Thickness measurements between two points (Eqs. 4.10and 4.11, or 4.1) exhibit a non-linear sensitivity to error at low angles between the dipvector and the measurement orientation, leading to a high probability of an artificiallyhigh average from multiple measurements. Smoothing of the attitude errors by struc-ture contouring leads to a better average thickness.

Where the thickness is known accurately from a complete exposure or from well-defined contacts in a borehole, the structure contours or bedding attitudes might beadjusted to conform to the thicknesses. The thickness measured between structurecontours is the best approach at the map scale where there is uncertainty in the data.

4.2Thickness of Folded Beds

In a folded bed, the dips of the upper and lower contact are not the same and the previousthickness equations are inappropriate. The fold is likely to approach either the planar dipdomain or the circular arc form. Equations for both forms are given in the next two sec-tions. For both methods it is assumed that the thickness is constant between the measure-ment points and that the line of the thickness measurement and the bedding poles are allin the plane normal to the fold axis. The latter condition is satisfied if the directionsof both dips and the measurement direction are the same. If the geometry is morecomplex than this, then a cross section perpendicular to the fold axis should be con-structed to find the thickness and projection may be required, as discussed in Chap. 6.

4.2.1Circular-Arc Fold

The thickness of a bed that is folded into a circular arc (Fig. 4.10) can be found if thedip direction of the bed and the well or traverse line are coplanar. In this situation thebedding poles intersect at a point. Let ρ1 be the smaller angle between the well and thepole to bedding, thus always associated with the longer radius, r1. The thickness, t, is

t = r1 – r2 . (4.12)

From the law of sines:

r2 = (L sin ρ1) / sin γ , (4.13)

r1 = (L sin (180 – ρ2)) / sin γ , (4.14)

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where ρ2 and r2 =, respectively, the angle between the bed pole and the well and theradius associated with the larger angle, and γ = ρ2 – ρ2. Substitute Eqs. 4.13 and 4.14into 4.12 and replace sin (180 – ρ2) with sin ρ2 to obtain the thickness:

t = (L / sin γ) (sin ρ2 – sin ρ1) . (4.15)

A typical data set is shown in Fig. 4.11a. The cross section is in the dip direction.The angles between the line of measurement and the poles to bedding are determinedas well as the acute angle between the poles (Fig. 4.11b). From Eq. 4.15, the true thick-ness of the bed is 234 ft.

Fig. 4.10.Thickness of a bed folded intoa circular arc. The dip of thebed and the well are co-planar.C is the center of curvaturewhere the poles to beddingintersect

Fig. 4.11.Example of thickness determi-nation of a circularly foldedbed. a Field data: bedding dipsare shown by heavy lines, dis-tance between exposures ofupper and lower contacts is300 ft on a line that plunges 10°.b Angles required for the thick-ness calculation

4.2 · Thickness of Folded Beds

100 Chapter 4 · Thickness Measurements and Thickness Maps

4.2.2Dip-Domain Fold

The dip-domain method can be used to find or place bounds on the thickness of a unitthat changes dip from its upper to lower contact (Fig. 4.12). For constant bedding thick-ness, the axial surface bisects the angle of the bend. The total thickness of the bedalong the measurement direction, t, is the sum of the thickness in each domain, foundfrom Eq. 4.1 as

t = L1 cos ρ1 + (L – L1) cos ρ2 , (4.16)

where ρ1 = the angle between the well and the pole to the upper bedding plane, ρ2 = theangle between the well and the pole to bedding of the lower bedding plane, and L1 = theapparent thickness of the upper domain. If the position of the dip change can be lo-cated, for example with a dipmeter, then it is possible to specify L1 and find the truethickness. If the location of the axial surface is unknown, the range of possible thick-nesses is between the values given by setting L1 = 0 and L1 = L in Eq. 4.16. The circu-lar-arc thickness (Eq. 4.15) is usually half-way between the extremes that are possiblefor dip-domain folding.

4.3Thickness Maps

Thickness maps are valuable for both structural and stratigraphic interpretation pur-poses. Because multiple measures of thicknesses can be mapped, care is required inthe interpretation. The calculated thickness is related to the dip and so uncertainties

Fig. 4.12.Thickness of a dip-domainbed that changes dip in themeasured interval, in a crosssection normal to the fold axis

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or errors in the dip may appear as thickness anomalies. An isopach map is a map ofthe true thickness of the unit (t, Fig. 4.1) measured normal to the unit boundaries (Batesand Jackson 1987). An isocore map is defined as a map of the vertical thickness of aunit (tv; Fig. 4.1; Bates and Jackson 1987). The drilled thickness in a deviated well(ts, Fig. 4.1) will usually differ from either the true thickness or the vertical thickness.It is not possible to correct the thickness in a deviated well to the vertical thickness orto the true thickness without knowing the dip of the bed. The thickness differencesresulting from the different measurement directions are not large for nearly horizon-tal beds cut by nearly vertical wells, but increase significantly as the relationships de-parts from this condition. The effects of stratigraphic and dip variations on thicknessmaps are considered here. The effect of faults on isopach maps are discussed in Sect. 8.5.

4.3.1Isopach Maps

An isopach map is used to show thickness trends from measurements at isolated points(Fig. 4.13a). An isopach map can be interpreted as a paleotopographic map if the up-per surface of the unit was close to horizontal at the end of deposition. If the paleo-topography was controlled by structure, then it can be considered to be a paleostructuremap. The thickness variations represent the structure at the base of the unit as it wasat the end of deposition of the unit. The trend of increased thickness down the centerof the map in Fig. 4.13a could imply a filled paleovalley.

The slope of the base of the paleovalley can be determined from the thickness dif-ference and the spacing between the contours according to the geometry of Fig. 4.13b:

δ = arctan (∆t / h) , (4.17)

where δ = the slope, ∆t = the difference in thickness between two contours, and h = thehorizontal (map) distance between the contours, measured perpendicular to the con-tours. For the map in Fig. 4.13a, the slope implied for the western side of the paleovalleyis about 0.5° (∆t = 10, h ≈ 900). Stratigraphic thickness variations could be caused bygrowing structures. The dip calculated from an isopach map using Eq. 4.17 could rep-resent the structural dip that developed during deposition. According to this interpre-tation, Fig. 4.13a could represent a depositional syncline.

Fig. 4.13. Paleoslope from thickness change. a Isopach map. Dots are measurement points; h is the lo-cation of a cross section. b Cross section perpendicular to the trend of the thickness contours, inter-preted as if the upper surface of the unit were horizontal

4.3 · Thickness Maps

102 Chapter 4 · Thickness Measurements and Thickness Maps

As an example, an isopach map is constructed from the data in Fig. 4.14a. For thepurpose of discussion, the points are contoured by both triangulation (Fig. 4.14b) andkriging (Fig. 4.14c). The paleogeographic implications of the map should be consid-ered before either map is accepted. The triangulated map suggests a stream channelwhereas the kriged map suggests an isolated depocenter. Both computer contouringmethods close the contours within the map area. Re-examination of the data revealsthat if the unit represents a channel, it could be extended off the map to both the northand south and still be consistent with all the control points. Accepting the depocenterinterpretation, it can be visualized in 3-D as a paleostructure map by reversing the signon the contours so that the thickest part plots as the deepest (Fig. 4.14d).

Tthickness trends on isopach maps could alternatively represent unrecognized faultsthat are too small to be identified directly. A normal fault will cause a thinning of theisopachs and a reverse fault will cause a thickening. Section 8.5 discusses faults onisopach maps. Figure 4.14 could represent a reverse fault that is too small to repeat thetop and base of the unit.

4.3.2Isocore Maps

Isocore maps are particularly valuable for determining the volume of a unit present inthe area of interest. The area enclosed by each isocore contour is multiplied by the

Fig. 4.14. Interpretation of a thickness map. a Thickness data. b Triangulation contouring. c Krigedmap on a 10 × 10 grid. d Paleostructural interpretation produced by making the thicknesses negativeon the kriged map. The plane of zero elevation is shown above the map and would represent the uppersurface of the unit at the end of deposition

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contour interval and then summed to obtain the volume. This is only an approxima-tion because it assumes that the volume consists of a stack of vertical-sided regions.The smaller the contour interval, the better the estimate.

Apparent thickness variations in vertical wells can provide a very sensitive tool forstructural analysis if bed thickness is known (Fig. 4.15). True dip is not known in awell unless the interval of interest has been cored or a dipmeter log is available. If thetrue thickness is known, the dip can be determined by solving Eq. 4.1 to obtain

δ = arccos (t / tv) . (4.18)

If a unit has a true thickness of 100 m, a dip of 10° gives an exaggerated thicknessof 102 m, 20° gives 106 m, 30° gives 115 m, 40° gives 131 m, 50° gives 156 m. The impor-tance of this effect will depend on the level of detail being interpreted, but will becomesignificant for nearly any purpose at dips over 20–30°. For example, a measured thick-ness of 103 ft for a unit having a true thickness of 100 ft may be stratigraphically in-significant, yet implies a dip of 14° which is steeper than the dip producing the closurein many oil fields. If the unit mapped in Fig. 4.13a actually has a constant thickness of100 units, then the dips in the center of the map where the isocore thickness is 110 unitsmust be 25°. Alternatively, the unit could be horizontal and the wells in which the thick-nesses were observed could deviate 25° from the vertical.

As an example of the importance of dip on the variation of apparent thickness, thethickness map in Fig. 4.14a is reinterpreted as representing isocore thicknesses of afolded unit of constant stratigraphic thickness. The thickness variations are convertedinto dips with Eq. 4.18, assuming a stratigraphic thickness of 150 units, and the valuestriangulated (Fig. 4.16). What was previously interpreted as a thickness trend is nowseen as a dip trend with dips up to 34°. This could represent a significantly folded unitwith the steepest dips representing the inflection point on the limb between synclineand anticline. A structure contour map of the unit should be constructed and exam-ined for correspondence between the trends. Note that thicknesses smaller than theassumed constant value yield spurious values when processed with Eq. 4.18. The north-east and southwest corners of the map in Fig. 4.16 would be better interpreted as re-gions of zero dip because the thicknesses are close to and slightly less than the as-sumed regional constant value.

Fig. 4.15.Dip of bed related to verticalapparent thickness, tv: verticalthickness; t: true thickness,δ : dip of bed

4.3 · Thickness Maps

104 Chapter 4 · Thickness Measurements and Thickness Maps

4.4Derivation: Map-Angle Thickness Equations

The traditional method for determining thickness based on data from a geologic map ona topographic base results in two equations, depending on the relative dip of topographyand bedding. The following derivations are after Dennison (1968). If the ground slopeand the dip are in the same general direction, Fig. 4.17 shows that t = ah = the true thick-ness, bc = fe = v = the vertical elevation change, ac = h = the horizontal distance from theupper to the lower contact, angle cae = α = the angle between the measurement directionand the true dip, and angle aej = feg = δ = the true dip. The thickness is

t = aj – hj = aj – eg , (4.19)

where

eg = v cos δ , (4.20a)

aj = ea sin δ , (4.20b)

ea = h cos α . (4.20c)

Substitute Eqs. 4.20 into 4.19 to obtain

t = |h cos α sin δ – v cos δ | . (4.21)

Fig. 4.16.Dip map of data from Fig. 4.14ainterpreted as isocore thick-nesses measured in a folded,constant-thickness unit. Truestratigraphic thickness is 150,dips in degrees

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If the dip of bedding is less than the dip of the topography, the second term in Eq. 4.21(D2.24) is larger than the first, giving the correct, but negative, thickness. Taking theabsolute value corrects this problem.

The thickness of a unit which dips opposite to the slope of topography (Fig. 4.18) is

t = eg + eh , (4.22)

where

eh = ea sin δ . (4.23)

Substituting Eqs. 4.20a, 4.20c, and 4.23 into 4.22:

t = h cos α sin δ + v cos δ . (4.24)

Fig. 4.17.Thickness parameters for abed and topographic surfacedipping in the same generaldirection

Fig. 4.18.Thickness parameters for abed and topographic surfacedipping in opposite generaldirections

4.4 · Derivation: Map-Angle Thickness Equations

106 Chapter 4 · Thickness Measurements and Thickness Maps

4.5Exercises

4.5.1Interpretation of Thickness in a Well

Based on the data in Table 2.2, what is the isocore thickness of the Smackover? Whatis the true thickness of the Smackover given its attitude of 12, 056 from the dipmeterlog and the orientation of the well from Exercise 2.9.1? Discuss the significance of thedifference between the isopach and isocore thickness.

4.5.2Thickness

Given a bed with dip vector 10, 290, and a measured thickness of 75 m in a verticalwell, use the universal-thickness equation to determine its true thickness.

4.5.3Thickness from Map

Use the map of the Blount Springs area (Fig. 2.27) to answer the following ques-tions. What is the thickness of the Mpm between the structure contours using themap-angle equations and the pole-thickness equation? Are the results the same? Ifthey are different, discuss which answer is better. What is the difference betweenthe true thickness and the vertical thickness of the Mpm? What is the thickness ofthe Mpm in its northeastern outcrop belt, assuming that the dip is 28° at itsnorthwestern contact and the value determined above occurs at its southeasterncontact? Use the concentric fold model and the dip-domain model. Discuss theeffect of changing the location of the axial surface on the thickness computedwith the dip-domain model. Measure the thickness of the Mpm at 5–10 loca-tions evenly distributed across the map. Measure thicknesses between structurecontours where possible. Construct an isopach map from your thickness measure-ments. Is the unit constant in thickness? What would be the apparent thickness ofthe Mpm in a north-south, vertical-sided roadcut through the northwestern limbof the anticline?

4.5.4Isopach Map

Make an isopach map of the sandstone thicknesses on the map of Fig. 4.19. Thethickest measurements form a trend that could be a channel or the limb of amonocline. If the thickness anomaly is due to a dip change, what is the amount?If the thickness anomaly is due to paleotopography, what is the maximum topographicslope?

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Fig. 4.19. Map of thicknesses (in feet) in the John S sandstone

4.5 · Exercises